Aggregating infinite utility streams with inter-generational equity

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Aggregating

Infinite

Utility

Streams

with

Inter-generational

Equity

Kaushik

Basu

Tapan

Mitra

Working

Paper 02-19

April

26,

2002

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Aggregating

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Utility

Streams

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Inter-generational

Equity

Kaushik

Basu

Tapan

Mitra

Working

Paper 02-19

April

26,

2002

Room

E52-

251

50

Memorial

Drive

Cambridge,

MA

02142

This

paper can

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^^i^ss""''

Aggregating

Infinite

Utility

Streams

with

Inter-generational

Equity*

Kaushik Basu^and

Tapan

Mitra^

April

26,

2002

Abstract

It has been

known

that, in aggregating infinite utility streams, there does not exist

any social welfare function, which satisfies the axioms ofPareto, inter-generational equity

and continuity.

We

show

that the impossibility result persists even without imposing the

continuityaxiom. Hence, theproblemof

accommodating

inter-generational equityis

more

obstinatethan previously supposed.

The

paper goes

on

to explore thescope for obtaining

possibility results by weakening the Pareto axiom and placing restrictions on the

domain

ofutilities.

Journalof

Economic

Literature Classification Nmnbers: DOO, D70, D90.

*Theauthors aregrateful to Jorgen Weibullfor helpful suggestions.

^Department ofEconomics, M.I.T., Cambridge,

MA

02142, Email: [email protected] and Department of Eco-nomics, Cornell University, Ithaca,

NY

14853.

1

Introduction

The

subject ofinter-generational equity in the context of aggregating infinite utility streams has

been

of enduring interest to economists.

Ramsey

(1928)

had

maintained that discoimting one

generation's utihty or

income

vis-a-vis another's to

be

"ethically indefensible",

and

something

that "arises merely

from

the

weakness

of the imagination". Since it is generally agreed that

any

such process of aggregation should satisfy the Pareto principle,

Ramsey's

observation raises

the question of

whether one

can consistently evaluate infinite utility streams while respecting

intergenerational equity,

and

at least

some

form

ofthe Pareto axiom.

In

an

important contribution tothis literature, Svensson (1980) established the general

possi-biUty result (for asocial welfare relation

(SWR))

that one can find

an

ordering, that satisfiesthe

axioms

ofPareto

and

inter-generationalequity. It is

worth

noting here that

he

obtains the

(com-plete) ordering

by

non-constructivemethods; specifically, hedefinesa partialorder satisfyingthe

two

axioms,

and

then completes the order

by

appealing to Szpilrajn's

lemma.

Svensson's paper builds

on

the earlier seminal contribution to this Hterature

by

Diamond

(1965). In this paper,

Diamond

presents the celebrated theorem^ that there does not exist

any

social welfare fimction

(SWF)

(that is, a function that aggregates

an

infinite

stream

into a real

nimiber) satisfying three axioms: Pareto, intergenerational equity

and

continuity (in the sup

metric).

Neither contribution addresses the following question: does there exist a social welfare

func-tionsatisfying intergenerational equity,

and

the Pareto

axiom?

Since continuity of the

SWF

in

Diamond's

exercise isa technical

axiom

(incontrast to the other

two

axioms),

we

think that this

is

an

issue

worth

investigating.''

The

principal task of this paper is to provide

an

answer to this

question.

The main

result that motivates this paper is the finding that the continuity

axiom

is

not needed for the

Diamond-

Yaari impossibihty theorem.

If

we

denote

by

Y

(asubset ofthereals) theset ofadmissibleutilitylevelsofeach generation,

and

by

X

the set of infinite streams of these utility levels,

an

SWF

is a function

from

X

to the

reals. It turns out that the answer to the question

posed

in the previous

paragraph depends on

the nature of the utility space (that is, the set, denoted

by

y

),

and on

the

form

ofthe Pareto

axiom.

As

a consequence, our resultscan

be

classified conveniently into fourparts.

First, if one adopts the standard

form

of the Pareto

Axiom,

then regardless ofthe specific

nature of the utility space, one obtains the general impossibihty

theorem mentioned

above:

there is

no

SWF

which

satisfies the Pareto

and

equity axioms. In other word,

any

Paretian

SWF

is necessarily inequitable. In particular, this

means

that the complete orderings, satisfying

the Pareto

and

equity axioms, obtained

by

Svensson,

do

not have real-valued representations.

Indeed, Svensson (1980) goes

on

to

show

that there are complete orderings,

which

satisfy the

Pareto

and

equity axioms,

and

in addition are continuous (ina topology

weaker

than

thediscrete

topology). Clearly, the continuity ofthe ordering (obtained

by

him) is unable to

overcome

the

representation problem.

The

proof of our result has

an

affinity with the demonstration that

lexicographic preferences

do

nothave areal-valuedrepresentation: one "runsoutofreal

numbers"

in a similar way.

Suppose

we

do want

to use a (real-valued)

SWF.

How

does one get

around

this

problem

of

"•in this connection, we

may

note that the Hne of research, initiated by Koopmans (1960), and continued

by Koopmans, Diamond and Wilhamson (1964) and Diamond (1965) estabhshes that Paretian social welfare

relations, continuous in suitably defined metrics, necessarily exhibit "impatience" in the sense that the current

generation receives more favorable treatment than future generations. Burness (1973) explores the impatience

respecting intergenerational equity?

One

way

to

do

this is to use weaker forms of the Pareto

axiom. If

we

use the

weak

Pareto axiom,

and

restrict the utihty space to be a subset of the

set of non-negative integers, then

one

can estabhsh a possibiHty result: there is a social welfare

fimction satisfying the equity

and

weak

Pareto axioms. Specifically, ifthe utility space consists

of just

two

elements (corresponding to, say, the "good"

and

"bad" states,

and

represented

by

1

and

respectively, so that

Y =

_{{0, 1}} ), the above

two

results

would

indicate that there exists

an

SWF

satisfying equity

and

the

weak

Pareto axiom, but thereis

no

SWF

satisfyingequity

and

the standard Pareto^ axiom.

Encouraged

by

our second result,

one might

ask

whether

it is always possible to obtain a

SWF

satisfyingthe equity

aad

weak

Pareto

axioms

(regardless ofthenature ofthe utilityspace,

Y

).

Our

third result indicates that the answer is in the negative. If

Y

is the closed interval

[0, 1] (the utility space chosen

by both

Diamond

(1965)

and

Svensson (1980) in estabhshing

their results,

mentioned

above),

one can

show

that there is

no

SWF

satisfying the equity

and

weak

Pareto axioms. This result is a generalization of

Diamond's

since it

shows

that one does

not

need

to

impose

the continuity

axiom

to get his impossibihty result.

One

also does not

need

the full strength of the Pareto axiom;

weak

Pareto (and equity) suffices. [Indeed, as

we

demonstrate, one does not even

need

the

weak

Pareto axiom; a

weaker form

ofit,

which

we

call

the

dominance

axiom, suffices].

Of

course, continuity is still at the heart ofthe demonstration of

this impossibility result.

The

interesting observation is that the extent of continuity

needed

for

the

Diamond-

Yaari technique to

work

is already implied

by

the

weak

Pareto

axiom

in the _{[0, 1]}

utihty space case,

making

a separate continuity

axiom

superfluous. [Specifically, the underlying

have

some

monotonicity properties,

and monotone

functions

on

[0, 1] are continuous almost

everywhere]

.

Finally, as

we

note in our fourth result, if

we

weaken

the Pareto

axiom

sufficiently to

what

we

call

"weak

dominance", a general possibility result can

be

obtained: there exists

an

SWF

satisfying the equity

and

weaJc

dominance

axioms, regardless of the nature ofthe utility space,

Y. It is

worth

remarkingthat,

when

the utihty space,

Y

, is [0, 1],

any

SWF

satisfyingthe equity

and

weak

dominance

axioms, obtained

by

applying our last result, necessarily violates the

weak

Pareto

axiom

(given our third result).

2

Paretian

Social

Welfare

Functions

are Inequitable

Let

R

be

the set of real numbers,

N

the set ofpositive integers,

and

M

the set of non-negative

integers.

Suppose

F

C

R

is theset ofallpossibleutilitiesthat

any

generation can achieve.

Then

X

=

Y^

is the set ofall possible utility streams. If {xt}

eX,

then {xt}

=

(xi,a;2,

),

where, for

all

teN,

xteY

representsthe

amount

ofutility thatthe t"' generation (that is, the generation of

period t) earns. For all

y,zeX,

we

write y

>

z \iyi

>

Zi, for allzeN;

we

write y

>

z iiy

>

z

and

y T^ z;

and

we

writey

>>

z, ii yi

>

Zi, for all

ieN.

If

Y

has only one element, then

X

is a singleton,

and

the

problem

of ranking or evaluating

infinite utihtystreamsis trivial. Thus, withoutfurther mention, theset

Y

willalwaysbe

assumed

to haveat least

two

distinct elements.

A

social welfare function

(SWF)

is a

mapping

Consider

now

some

of the gixioms that

we

may

want

the

SWF

to satisfy.

The

first

axiom

is

the standard Pareto condition.

Pareto

Axiom:

For all

x,yeX,iix>y,

then

W{x)

>

W{y).

The

next

axiom

is the

one

that captures the notion of 'inter-generational equity'.

We

shall

call it the 'anonymity axiom'.^ It is equivalent to the notion of'finite equitableness' (Svensson,

1980) or 'finite anonymity' (Basu, 1994).'^

Anonymity

Axiom:

For all

x,yeX,

ifthere exist

i,jeN

such that Xi

=

yj

and

Xj

=

yi,

and

for

A;eN

-

{i,j}, Xk

=

yk, then

W{x)

=

W{y).

The

principal result ofthis section is that there is

no

social welfare function

which

satisfies

both

axioms.

Theorem

1 There does not exist

any

SWF

satisfying the Pareto

and

Anonymity

Aodoms.

Proof. Assume, on

the contrary, that thereis asocial welfare function,

W

:

X

—

>R,

which

satisfies thePareto

and

Anonymity

Axioms.

Since

V

C

M

containsat least

two

distinct elements,

there exist real

numbers

y°

and

y^ in Y, withy^

_>

y".

Without

loss ofgenerality,

and

toease the

writing,

we

may

suppose that y^

=

1

and

y'^

=

_0.

Let

Z

denotethe

open

interval _{(0, 1),}

and

let ri, r2, r^, ...

be an

enumeriationofthe rational

numbers

in Z. Let

_{p be an}

arbitrary

number

in Z.

We

define the set:

E{p)

=

{n

G

N

: r„

<

p}

.

Ininformal discussionsthroughout thepaper, the terms "equity" and "anonymity" areused interchangeably.

^The Anonymity Axiomfiguresprominently inthesocialchoicetheoryliterature, whereitisstatedas follows:

thesocialorderingisinvariant tothe information regardingindividualorderings as to

who

holdswhichpreference

ordering. Thus, interchanging individual preference profiles does not change the social preference profile. See

Clearly, E{p) is an infinite set.

We

now

define a sequence a{p)

=

(a(p)i, 0(^)2, ••) as follows:

a{p)n

=

<

1

ifneEip)

otherwise

Note

that the sequencewill have

an

infinite

number

of I's

and an

infinite

number

ofO's. Define

the sequence b{p)

=

(fe(p)i, 6(^)2,...) as the

same

as the sequence a{p) except that the first

appearing in the a(p)sequence is replaced

by

a 1.

By

the Pareto axiom,

we

have W{b{p))

>

W{a{p)).

We

denote the closed interval [W{a{p)), W{b{p))]

by

I{p).

Now,

let qbe

an

arbitrary real

number

in Z, withq

>

p- Clearly,

we

must

have

E{p)

C

E{q),

for if

n

e

E{p), then r„

<

p,

and

since

p

<

q,

we

must

have r„

<

q, so that

n

G

E{q). Further,

there are

an

infinite

number

of rational

numbers

in the interval {p,q)- Thus,

comparing

the

sequence a{p) with the sequence a{q),

we

note that:

(i)

i/n

e

N,

and

a(p)„

=

1, then a(g)„

=

1 ₍₁₎

and:

(n) there are

an

infinite

number

0/

n

£

N

for which

a(p)„

=

and

a(g)„

=

1 ₍₂₎

We

now

proceed to

compare

the sequence a{q) with the sequence b{p). Let

m

G

N

be

the

indexforwhichthe seqencea{p) differs

from

thesequenceb{p); that is, a(p)m

=

0,

and

b{p)m

=

a{q)

>

b{p),

and

so:

W{a{q))

>

W{b{p)) (3)

In case (B),

we

proceed as follows. Let

M

be the smallest integer for

_{which a{p)M}

—

while a{q)M

=

1-

By

observation (2) above, such

an

M

exists. Also, clearly

M

^

m,

for

a{q)M

=

1 while a{q)m

=

0. Since b{p) differs from a{p) in only the index

m,

we

_{have 6(p)m}

=

0.

Now,

define b'{p) as follows: _6'(p)m

=

0, b'{p)M

=

1, b'{p)n

=

b{p)n for all

n

G

N

such that

n

_^

m,n^

M.

Since b{p)m

=

1,

and

_6(p)m

=

0, the

Anonymity

axiom

implies that:

W{b'{p))

=

W{b{p)) (4)

Comparing

b'{p) with a{q),

we

note that b'{p)m

=

=

a(g)m, b'{p)M

=

1

=

a{q)M,

and

for aU

n

G

N

such that

n

y^

m,n

^

M,

we

have 6'(p)„

=

b{p)n

=

a(p)„

<

a(g)„

by

observation (1).

By

observation (2),

we

must

therefore have a{q)

>

b'(j)), so that

by

the Pareto

Axiom:

W{a{q))

>

W{b'{p)) (5)

Combining

(15)

and

(16),

we

get:

Wia{q))

>

W{b{p))

(6)

Thus, in

both

cases (A)

and

(B),

we

have

W{a{q))

>

W{b{p)). This

means

that the interval

/(g)

=

[W{a{q)), W{b{q))] is disjoint

from

the interval [W{a{p)), W{b{p))], the latter interval

lying entirely to the left ofthe former interval

on

the real line.

(0,1) are non-overlapping. But, this

means

that toeachreal

number

p G

(0, 1),

we

canassociate

a distinct rational

number

(in the interval /(p)), contradicting the countability of the set of

rational numbers.

Remark:

The

construction (used in the

above

proof) of

an

uncountable family ofdistinct nested sets

E{p), with eachset containing

an

infinite

number

of positive integers, can

be

found in Sierpinski

(1965, p. 82).

Let us

now

suppose that

we

abandon

the search for

an

SWF

and

instead look for a social

welfare relation

(SWR),

We

then have the result

due

to Svensson (1980) that there is an

SWR

which

satisfies the (appropriate relational versions of the) Pareto

and

Anonymity

axioms. For

reasons of completeness

we

briefly review Svensson's result.

We

do

this because, the use of a variant of Szpilrajn's

Theorem

(due to

Suzumura,

1983) allows us to give a particularly easy

proofofit. Also, Svensson (1980) restricts his exercise to the case

where

Y

is the closed interval

[0, 1];

we

state the version of his result

which

apphes to any utihty space y.His proof, as well as

ours, applies to this

more

general setting.

Formally,

an

SWR

is a binary relation,

_{^, on}

X,

which

is complete

and

transitive.

We

use

>-

and

~

to denote, respectively, the

asymmetric

and symmetric

parts of ^.

The

properties of

Paxeto

and

Anonymity

for

an

SWR

are easyto define.

We

shall call these

axioms

^-Pareto

and

^-Anonymity

to distinguish

them

from

the

axioms

applied to

an

SWF.

^-Pareto

Axiom:

For all x,

ye

X

, x

>

y implies x >- _y.

^-Anonymity

Axiom:

For all x,y

eX,

ifthere exist i,jeN, suchthat Xj

=

yj

and

Xj

=

yt

and

First, let us give a statement of

Suzumura's

theorem. Let il

be

a set ofalternatives. If

R

is

a

binary relation

on

Q

and R* an

ordering

on

fi,

we

shall say that

R*

is

an

ordering extension

of i? if, for all x,y

eO,

a; /f!t/ implies

xR*

y.

We

say that

R

is consistent if, for all

teN,

and

for

all x^, x^,...,x*eQ, [x^Rx"^

and

not x^Rx^,

and

for all A;e{2, 3,...,t

—

1}, x^Rx'^^^] implies not

x^RxK

Lemma

1 Szpilrajn's Corollary [Suzumura, 1983,

Theorem

A

(5)]:

A

binary relation

R

on

fl

has

an

ordering extension if

and

only ifit is consistent.

In contrast to

Theorem

1,

we

now

have:

Theorem

2

(Svensson, 1980)

; There exists a social welfare relation satisfying the

^-Pareto

and "^-Anonymity

Axioms.

Proof.

Define

two

binary relations,

P

and

/,

on

X,

asfollows. For allx,y

eX,

x

>

y implies

xPy.

And

ifthere exists i,j such that Xi

=

yj

and

Xj

=

yt

and

Xk

=

Vk, for all k

^

i,j, then

xly.

Now

define the binary relation

R

as follows:

xRy

<^

xPy

or xly.

It iseasy to verify that

R

is consistent. Hence,

by

Szpilrajn's Corollary, it has

an

ordering

extension ^. Clearly

^

satisfies the

^-Pareto

Axiom

and

the

^-Anonymity

Axiom.

Remcirks:

(i)Theorem2

shows

thatthereis

no

inherent conflict

between

the conceptofintergenerational

equity

and

the Pareto principle.

Theorem

1

shows

that a conflict arises

when

we

try to obtain

an

evaluation of utility streams in

terms

ofreal numbers, while respecting these

two

properties.

(ii)

A

simple

example

of our set-up is one

where

the utility space,

Y

=

{0, l}.One might

interpret this as follows: there are precisely

two

states in

which

eachgeneration might find itself,

a

good

state

and

a

bad

state.

The

utility obtained

by

eachgeneration is 1 inthe

good

state,

and

in the

bad

state.

Theorem

1 tells us that even in this simple set-up, there is

no

SWF

which

respects

Anonymity and

the Pareto

Axiom.

3

Relaxing

Pareto

If

we

want

to

work

with a social welfare function

and

respect inter-generational equity or

anonymity,

what

possibilities

do

we

have? In the light ofour

Theorem

1,

what

we

explore in

this section is to relax the Pareto axiom.

It is arguable that for certain philosophical

and

even policy purposes

we

do

not need the

full-brunt of the Pareto condition (even if

we

are

committed

Paretians) simply because all the

possibilities that are technically allowed in our specification ofthe

domain,

may

not arise

under

any eventuaUty. Indeed for certain ethical discourses involving the comparison of the

moral

worth

of individual actions

and

universalizable rules (see Basu, 1994) it

may

be

enough

to

be

armed

with the following

weak

Paretianism.*^

Wccik Pareto

Axiom:

For all x,_ye

X,

ifthere exists je

N

such that Xj

>

_yj, and, forall k _{y^ j,}

Xk

=

Vk, then

W{x)

>

W{y).

For all x,

yeX,ii

x

>>

y, then

W{x)

>

W{y).

Note

that if

we

recognize that

human

perception or cognition is not endlessly fine, so that

sufficiently small changes in well-being go imperceived, it

seems

reasonableto suppose that the

set of feasible utilities will

be

a discrete set.^

The same

is true if the benefits axe

measured

^Our

Weak

Pareo Axiom is somewhat different from the

Weak

Pareto Axiom used in social choice theory,

wheretypicallyit isstated as follows: ifevery individual ina society isbetteroif, thensociety isbetter off. See,

forexample, Sen(1977). Sometimes, though, inthe socialchoicetheory literature, this axiomiscalledthe Pareto

Axiom, andour ParetoAxiomiscalledthe Strong ParetoAxiom. See, forexample, Arrow(1963) andSen(1969).

*The limitsonhuman perceptionhave beenused inadifferentcontext byArmstrong(1939) toarguethatit is

implausible tosupposethat indifferenceis atransitive relation. For adiscussionof this issue in individualchoice

in

money

and

there is a well-defined smallest unit, as is true for all currencies (Segerberg

and

Akademi,

1976). Thus, it

seems

worthwhile to explore

whether

with

y

C

M

(which captures

this very reasonable possibility), there is asocial welfare function (on

X

) respecting

Anonymity

and

the Weeik Pareto Axioms.^

Our

next

theorem

provides

an

interesting possibility result.

Theorem

3

Assume

y

C

M

. There exists

an

SWF

satisfying the

Weak

Pareto

and

Anonymity

Axioms.

Proof.

For each

xeX,

let

E{x)

=

{yeX

: there is

some

N

eN,

such that yk

=

Xk for all

A;eN,

which

are

>

N}.

Let

3

be

the collection

{E

:

E

=

E{x)

for

some

xeX}.

Then

^

is a

partition of

X.

That

is, if

E

and

F

belong to $5, then either

E

=

F, oi

E

is disjoint

from

F;

further,

_UecqE

=

X.

Define a function,

_/

:

X

—

*•

M

as follows.

Given

any

x

eX,

_{let f{x)}

=

min{xi,2:2,...}. Since

XjcM

for all

ieN,

the set {a;i,X2,...} is a

non-empty

subset of the set of non-negative integers

and

therefore has a smallest element [Munkres, 1975, p. 32]. Thus, / is well-defined.

By

the

axiom

of choice, there is a function, g :

Q

^^

X

, such that

g{E)

e

E

for each

E

eQ.

Given any

xeX,

we

can

denote for each

N

>

1, {xi, ...,xn)

by x{N),

and

(xi -I- ... -I-

x^) by

I{x{N)).

Now,

given

any

x,t/in

£^e^,

define /i(x,y) =\imj^^oo[I{x{N))

—

I{y{N))]. Noticethat

h

iswell-defined,sincegiven

any

x,yin

E

eQ,

thereis

some

M

eN, suchthat [I{x{N))

—

I {y{N))]

is

a

constant for all A^

> M.

Now,

given

any

x,y in Ee'ii, define

H{x,y)

=

0.5[h{x,y)/[l

+

\h{x,y)\]].

Then

i/(x,y)e(-0.5,0.5).

theory, see Majumdar (1962).

^_{While our} choice of

F

as a subset of the set of non-negative integers is motivated by the imprecision of

human

perception, the mathematical technique usedtoobtainourpossibility resultapplies also tothecasewhere

Y

=

{(1/n) : n€ N},where clearly

human

perceptionhas to beconsidered tobe sufficientlyrefined.

We

now

define

VF

: J'C

^

M

as follows.

Given

any

xeX,

we

associate with it its equivalence

class, E{x).

Then,

using the fimction g,

we

get

g{E{x))e

E{x). Next, using the functions, h

and

H,

we

obtain h{x,g{E{x)))

and

H{x,g{E{x))).

Now,

define

W{x)

=

f{x)

+

H{x,g{E{x))).

The Anonymity

Axiom

can

be

verified as follows. Ifx,y are in

X,

and

there exist i,j in N,

such that Xi

—

yj

and

Xj

=

yi, while Xk

=

yk for _al^Ai^N, such that k

_^

i,j, then

E{x)

=

E{y). Furthermore, denoting this

common

set

by

E,

we

see that h{x,g{E))

=

h{y,g{E)),

and

so

H{x,g{E))

=

H{y,g{E)). Further, the set {xi,X2, ...} is the

same

asthe set _{yi,y2, •••}, so that

f(x)

=

f{y). Thus,

we

obtain:

W{x)

=

W{y).

The

Weak

Pareto

Axiom

can

be

verified as follows. If x,y are in

X,

and

there exists

ieN,

such that Xi

>

yi, while Xk

=

yk for all

keN,

such that k

_^

i, then

E{x)

=

E{y).

Furthermore, denotingthe

common

set

by

E,

we

see that h{x,g{E))

>

h{y,g{E)). Thisimplies

H{x,

g{E))

>

H{y,

g{E)). Further, the smallest elementoftheset {xi, X2, ...} is at least aslarge

as thesmallest elementofthe _{set {yi,}2/2,•••}, so that

we

have f{x)

>

f{y). Thus,

we

obtain the

desired inequality:

W{x)

>

W{y).

If

x,yeX,

and x

»

y, then

E{x)

7^ E{y). Thus,

we

will not

be

able to

compare

H{x,g{E{X)))

with H{y,g{E{y))). However,

we

do

know

that

H{x,g{E{x)))

>

-0.5,

and

H{y,g{E(y)))

<

0.5. Further, since

x

>>

y,

we

have f{x)

>

f{y)

+

1. Thus,

we

obtain:

W{x)

=

fix)

+

H{x,g{E))

>

f{y)

+

1

-

0.5

>

_/(y)

+

H{y,g{E))

=

W{y).

U

Remarks:

(i)

The

important point to note is that our

Weak

Pareto

Axiom demands

that the

SWF

be

positively sensitiveto

an

increaseinutilityofasinglegeneration,the utilitiesofother generations

being

unchanged

(and thereforethat it

be

positively sensitive to increases in utilitiesof

any

finite

number

of generations, the utilities of other generations being

unchanged

),

and

also that the

SWF

be positively sensitive to

an

increase in utilities of all generations. However, it need not

be

positively sensitive to

an

increase in utiUties of

an

infinite

number

of generations,

when

the

utihties of a (non-empty) set of generations is unchanged. This is the principal difference ofour

Weak

Pareto

Axiom

from our Pareto

Axiom.

(ii) Consider the simple set-up, introduced in the previous section,

where

Y

=

{0,1}.

Now,

Theorem

1 implies that there is

no

SWF

respecting the Pareto

and

Anonymity

Axioms.

And,

Theorem

3 implies that thereis

an

SWF

satisfying the

Weak

Pareto

and

Anonymity

Axioms.

It

followsthat for

any

social welfare function,

W,

so obtained, it

must

be the case that there exist

alternatives

x,y E

X

such that x

>

y,but

W{x)

<

W{y).

4

Generalizing

the

Diamond-

Yaari Result

The

possibility result ofthe previous section is reason for optimism,

and

we

might

ask

whether

it can

be

extendedto all utility spaces, Y.

We

show

in thissection that such

an

extension is not

possible

by

demonstratingthat

when

Y

isthe closedinterval [0,l],thenthereis

no

SWF

satisfying

the

Anonymity

and

Weak

Pareto axioms. This generalizes the result of

Diamond

(1965) in

two

respects.

Diamond

had

shown

that

(when

the utility space

Y

is [0,1]), there is

no

SWF

satisfying

Anonymity,

Pareto,

and

a continuity

axiom

(in the sup metric

on

X).

Owe

result

shows

that

the continuity

axiom

is redundant for this impossibility result. In addition, it

shows

that the

full

power

ofthe Pareto

axiom

is not

needed

for the impossibility result; the weaJiPareto cixiom

suffices for this purpose.

Theorem

4 Assume

Y =

[0, 1]. There does not exist any

SWF

satisfying the

Weak

Pareto

and

the

Anonymity

Axioms.

Instead ofprovingthis

theorem

directly,

we

shall establisha

more

powerful theorem, ofwhich r

Theorem

4 is

an

obvious corollary.

To

prove this stronger result, let us next define a very

weak

form

ofthe Pareto criterion,

which

we

call the

Dominance

Axiom.

Dominance

Axiom:

For all x,y

eX,

if there exists je

N

such that Xj

>

_yj, and, for all k

^

_j, Xk

=

Uk, then

W{x) >

W{y).

For all

x,yeX,

ifx

>>

y, then

W{x)

>

W{y).

Note

that the last inequality in the statement of this

axiom

is a

weak

inequality, unlike in

the definition of the

Weak

Pareto

Axiom.

Hence,

Weak

Pareto is stronger than

Dominance.

It is not as if

we

wish to

recommend

the use of such a

weak form

of the Pareto condition, but

since

we

are going to prove

an

impossibihty result, clearly it is better to use as

weak

an axiom

as one can. Further, our proof indicatesthat it is preciselythe

Dominance

axiom

that is needed

to obtain the impossibility result.

Theorem

5

Assume

Y =

[0,1]. There does not exist

any

SWF

satisfying the

Dominance

Axiom

and

the

Anonymity

Axiom.

Proof.

To

establish the theorem,

assume

Y

=

[0, 1]

and

that there exists a social welfare

function,

W

:

X

^M,

which

satisfies the

Dominance

and

Anonymity

Axioms.

Denote

the vector (1,1, 1,...) in

X

by

e. Define the sequence

u

in

X

as follows:

u

=

(1, 1,0,1/2, 1,0, 1/4,2/4, 3/4,1, ...) (7)

This sequence isbest understood assequenceu, defined below, with the first

term

changed from

to 1. For se7

=

(-0.5,0.5), define:

x(5)

=

0.512

+

0.25(1

+

s)e (8)

Then

(l/8)e

<

x{s)

<

(7/8)e,

and

so x{s)

eX

for each sel. Define the function,

/

: 7

^

R

by:

f{s)

=

W{x{s))

By

the

Dominance

Axiom, /

is

monotonic

non-decreasingin s

on

7.

Thus

/

has onlya countable

number

of points ofdiscontinuity in 7. Let ae7 be a point of continuity ofthe function /.

Define the sequence ti in

X

as follows:

ti

=

(0, 1,0, 1/2, 1,0, 1/4, 2/4, 3/4, 1, ...) (9)

and

then define:

a;(a)

=

0.5w

+

0.25(1

+

a)e (10)

Clearly,

x{a)eX,

and

Xi{a)

>

Xi(a), while Xk{a)

=

Xk{a) for each

keN,

with k

_^

1. Thus,

by

the

Dominance

Axiom,

we

have:

W{x{a))

>

W{x{a))

We

denote [W{x{a))

-

W{x{a))]

by

9; then ^

>

0.

Denote max(0.5

—

a, 0.5

+

a)

by A;

then,

A

>

0. Since

/

is continuous at a, given the 9

defined above, there exists 6e (0,A), such that:

0<\s-a\<6

implies

_{\f{s)-f{a)\<9}

(11)

Note

that for

<

|s

—

a|

<

6,

we

always have sel.

We

define (following

Diamond),

for each

keN,

a sequence u''

by

starting with the sequence

u, interchanging the initial with the {k

+

l)st 1 appearing in «,

and

then interchanging the

sequence:

(l/2^2/2^...,(2'=-l)/2^o)

wuh

(o,l/2^2/2^...,(2'=-l)/2*=) ₍₁₂₎

so that:

u''

=

(1,1,0, 1/2,1,...,0, 0,

l/2^ 2/2^

...,(2^

-

1)/2^0, ...) (13)

Now,

for each A;eN,

we

use u^ to define x''{a) as follows:

x'=(a)

=

0.51*'=

+

_0.25(1

+

_{a)e (14)}

Clearly, x''(a)

eX

for each

keN.

Comparing

the expressions for

x

(a)

and

x''{a) in (10)

and

(14)

respectively,

and

usingthe expressions for

u

and

u'' _{in (9)}

and

(13) respectively,

we

see that the

Anonymity

Axiom

yields:

W{x''{a))

=

W{x{a))

for all

keN

₍₁₅₎

Choose

K

e

N

with

K

>2

such that (1/2^-2)

<

6,

and

define

S

=

{a

-

(1/2^-2)).

We

note that

<

{a

—

S)

<

6,

and

so

S

el, and:

W{x{S))

=

f{S)

>

f{a)

-9

=

W{x{a))

-

9 (16)

We

now

compare

the welfare levels associated with

x^{a)

and

x{S) as follows. Notice that:

x^{a)

=

0.5m^

+

0.25(1

+

a)e

=

0.5u

+

0.25(1

+

a)e

-

0.5(u

-

u^)

=

x(a)-0.5(u-u^)

>

x{a)

-

0.5(l/2^)e

=

0.512

+

0.25(1

+

a)e

-

0.5(l/2^)e

=

0.5u

+

0.25(1

+

a

-

{l/2^-'^))e

=

0.5u

+

0.25(1

+

a

-

{l/2^-^))e

+

0.25(1/2^-^)6

»0.5f2

+

0.25(1

+

5)e

=

x{S)

Thus,

by

the

Dominance

Axiom,

we

have:

W{x''{a))>W{x{S))

(17)

Using (15), (16)

and

(17),

and

the definition of6,

we

obtain:

W(x{a))

-9

=

W{x{a))

=

W{x^{a))

>

W{x{S))

>

W{x{a))

-

9

a contradiction which estabHshes our result.

The

intuitive idea behind our proof is straight-forward.

Diamond

had

used the

axioms

of

Pareto

and

Anonymity

in conjunction with another

axiom

concerning the continuity of

W

to

precipitate

an

impossibihty result.

Now,

as soon as

we

impose

the

Weak

Pareto

Axiom

(or the

Dominance

Axiom),

by

a standard result in analysis,

we

know

that

W

cannot

be

discontinuous everywhere. In other words,

W

must

havepointsof continuity.

And

we

show

that thatis

enough

to give us the impossibility result. In other words, there is

no

need to

assume

continuity.

The

minimal

amount

of continuity that is needed to create the impossibility arises naturally as a

consequence ofthe

Dominance

Axiom.

5

Weak

Domincince

What

we

have not yet found is

an

existence result

where

Y

=

[0, 1]

and

some

version of the

Pareto criterion is satisfied.

Theorem

5 indicates that

we

need to

weaken

the Paxeto criterion

further to get such a result. In fact

what

we

need is a condition in

which

only the first part of

the

Dominance

axiom

is asserted. If each policy question that

we

consider affects the utihties of

only afinite

number

ofgenerations, then this

"Weak

Dominance"

conditionsuffices in capturing

the idea of Pareto.

Weak

Dominance

Axiom:

For all

x,yeX,

if for

some

j

eN,

Xj

>

_y^, while, for all k

_^

_j,

Xk

=

yk, then

W{x)>W{y).

Theorem

6

There exists

an

SWF

satisfying the

Weak Dominance

and

Anonymity

Axioms.

Proof.

For each

xeX,

let

E{x)

=

{yeX

: there is

some

TVeN

, such that y^

=

Xk for all

keN,

which

are

>

A''}. Let $j

be

the collection

{E

:

E

=

E{x)

for

some

xeX}.

Then,

5

is a

partition of

X.

By

the

axiom

of choice, there is a function, 5 : Q'

—

>

X,

such that

g{E)

eE, for

each

EeQ.

Given

any

x,y in

E

e^,

define h{x,y)

=

lim;v—oo[-^(3^(-^))

—

^(y(-^))]-We

now

define

VT

:

X

—

>

R

as follows.

Given

any

x

e

X, we

associatewith itits equivalenceclass, E{x).

Then,

usingg,

we

get g{E{x))eE{x), and, using h,

we

obtain h{x,g{E{x))).

Now,

define

W{x)

=

h{x,g{E{x))).

The Anonymity

Axiom

and

the

Weak

Dominance

Axiom

are easily verified.

6

Conclusion

We

summarize

our results,

marking

out the terrain of

what

is possible

and

what

is not possible,

in

a

table:

[0,1]

N

Pareto _{Theorem^ 1^^}

Weak

Pareto Theorem4;: Theorem 3

Dominance Theorem'5 ':.

Weak

Dominance Theorem6

TABLE

1

It is

assumed

in Table 1 that the

Anonymity

Axiom

is satisfied.

The

rows represent a

weakening

sequence of Pareto-type axioms.

The

columns

represent the

domain

restrictions, to

wit,

what

Y

is equal to.

The

shaded area represents the zone of impossibihty (where

no

SWF

exists)

and

the

unshaded

areathe zoneof possibihty.

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